The first step in binocular stereopsis is to match features on the left retina with the correct features on the right retina, discarding false matches. The physiological processing of these signals starts in the primary visual cortex, where the binocular energy model has been a powerful framework for understanding the underlying computation. For this reason, it is often used when thinking about how binocular matching might be performed beyond striate cortex. This also led to the view that the solution to the stereo correspondence problem begins with a correlation-based computation (which the energy model implements). However, recent work by Doi et al. (2011, 2013, 2014) suggests that human observers can see depth in a class of stimuli where the mean binocular correlation is 0 (half-matched random dot stereograms). Half-matched random dot stereograms are made up of an equal number of correlated and anticorrelated dots, and the binocular energy model fails to signal disparity here. This has led to the proposition that a second, match-based computation must be extracting disparity in these stimuli. This description suggests that humans see depth in half-matched stereograms even though correlation-based V1 neurons do not signal disparity in their mean firing rate. This would be surprising as V1 activity is generally thought to be a necessary prerequisite for cyclopean depth perception, and would provide the first evidence that depth perception can occur without an explicit signal in V1. We therefore examined whether a simple modification to the binocular energy model - adding a point output nonlinearity - is sufficient to produce model cells that are disparity-tuned to half-matched random dot stereograms. Importantly, this nonlinearity has been proposed before as a possible explanation for one aspect of responses in real neurons that do not follow the energy model: anticorrelated stereograms produce attenuated responses. The modified model did show selectivity for disparity in halfmatched stereograms. Furthermore, a simple decision model based on these model cells reproduced psychometric functions generated by human observers, including reduced performance to large disparities and rapidly updating dot patterns. The model makes predictions about how performance should change with dot size in half-matched stereograms and temporal alternation in correlation. Measurements in human observers followed both of these predictions. Thus a single mechanism, based on already-known properties of V1 neurons, can account for the literature on half-matched random dot stereograms. This combination of modelling and human work cannot of course establish that real V1 neurons are able to signal the presence of disparity in these half-matched stereograms. We therefore recorded the activity of single neurons in V1 while awake monkeys maintained fixation. We found that they are indeed able to signal disparity in half-matched stimuli. The model we developed provided a good quantitative description of these data. This work reinstates the view that disparity-selective neurons in V1 provide the initial substrate for perception in dense cyclopean stimuli.